Summary: We introduce two properties: strong \(\mathrm{R}\)-property and \(C(q)\)-property, describing a special way of divergence of nearby trajectories for an abstract measure-preserving system. We show that systems satisfying the strong \(\mathrm{R}\)-property are disjoint (in the sense of Furstenberg) with systems satisfying the \(C(q)\) -property. Moreover, we show that if \(u_t\) is a unipotent flow on \(G/\Gamma\) with \(\Gamma\) irreducible, then \(u_t\) satisfies the \(C(q)\)-property provided that \(u_t\) is not of the form \(h_t\times \operatorname{id}\), where \(h_t\) is the classical horocycle flow. Finally, we show that the strong \(\mathrm{R}\)-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded-type Heisenberg nilflows.